Sharing voice memos with your computer

You can sync voice memos with
the primary iTunes library on your computer, then listen to memos on
your computer or sync them with another iPhone or iPod touch.

When you
delete a synced memo from iTunes, it stays on the device where it was
recorded, but is deleted from any other iPhone or iPod touch you
synced. If you delete a synced memo on iPhone, it's copied back to
iPhone the next time you sync with iTunes, but you can’t sync that
copy back to iTunes a second time.

Sync voice memos with iTunes:
Connect iPhone to your computer, then in iTunes select iPhone. Select
Music at the top of the screen (between Apps and Movies), select Sync
Music, select “Include voice memos,” and click Apply.

Voice memos
synced from iPhone to your computer appear in the Music list and in
the Voice Memos playlist in iTunes. Memos synced from your computer
appear in the Voice Memos app on iPhone, but not in the Music app

iTunes should actually sync Voice Memos if you have the option 'Sync Voice Memos' checked in the 'Music' tab when the iOS device is connected (either through USB or Wifi).

I actually forgot I had this item checked in iTunes, and noticed I had a new playlist named 'Voice Memos' in iTunes. These were rather short (only a minute to 5 long I think), but synced ok.

This answer by jmlumpkin is canonically correct, all existing memos sync over to a special "Voice Memos" playlist in iTunes. However, if you like to record voice memos that you will retrieve later on your computer, my suggestion would be to check out DropVox. Instead of having to go through iTunes, DropVox uploads your recordings to your DropBox folder as soon as you finish recording, which for me is far more convenient than syncing and dealing with iTunes. Personally, I have completely dropped Voice Memos and now completely use DropVox for everything.

Another avenue to avoid iTunes but still use the Voice Memo app is to use a software package like PhoneView to grab your existing memos directly from the device.

There is a free third-party utility that I find very useful in situations like this. Macroplant iExplorer, for Mac and Windows, lets you mount the file system of your iOS device in the Finder or Windows Explorer, so you can copy the voice memo files (and most other iOS document files) directly to your computer without using iTunes. Macroplant also provides other paid apps with additional functionality.

You can extract the voice memos from your iPhone back up by downloading this free application: http://www.iclarified.com/entry/index.php?enid=4974
This will allow you to read the files from your back up files. (saved my life)
The site also walks you through how to use it - it's really simple.
Once you have extracted the IOS files, go to the media folder and you will find recordings. That's where your voice memos will be!

Visit this article, which explains how to transfer the memo to iTunes.

Dock your iPhone via USB

Click the name of your iPhone under Devices in the left column

Click the Music tab

Click the checkbox for Include voice memos

Sync your device

Your files will appear under a new Voice Memos playlist

Then, if you want to have it play on Apple Tv or hear it from your phone and it still doesn't work, there is another trick.

Go into iTunes, hold the Ctrl button and click on the song/memo you've imported. Choose the "Create an AAC Version" option. It will make a copy of the song above or below the original in iTunes. Then you can play this on Apple TV or off your phone like a regular song.